Furthermore, in the practical implementation, the BS and IRS are generally installed at the high-altitude sites with few local scatters, which means that the BS-IRS link is more likely to form the line-of-sight (LoS) propagation link with high Rician factor and long channel coherence time. Therefore, it is reasonable to assume that the small-scale fading of the cascaded reflection channels is mainly determined by the IRS-user link, i,e., \({h_{k,{m_k}}} \sim \mathcal {CN}\left( {0,\beta _k^2} \right)\) , where \(\beta _k^2\)  denotes the path loss of the cascaded reflection channels via the \(k\) th IRS. Note that all the \({M_k}\)  elements in the \(k\) th IRS experience the same degree of path loss. As a result, for the \(m_k\) th reflecting element at the \(k\) th IRS, we have \(\mathbb {E}\left\lbrace  {{{\left| {{h_{k,{m_k}}}} \right|}^2}} \right\rbrace  = \beta _k^2\) . In addition, according to the conclusions of [1]}, it is not difficult to verify that, assuming \({h_{k,{m_k}}} \sim \mathcal {CN}\left( {0,\beta _k^2} \right)\)  and \({\varepsilon _{k,{m_k}}} \sim \mathcal {CN}\left( {0,\delta _{k,{m_k}}^2} \right)\)  for \(\forall k \in \mathcal {K},\ \forall {m_k} \in {\mathcal {M}_k}\) , we have
\(\mathbb {E}\left\lbrace  {\mathcal {R}\left\lbrace  {\frac{{\left( {h_{k,{m_k}}^* + \varepsilon _{k,{m_k}}^*} \right){h_{k,{m_k}}}}}{{\left| {{h_{k,{m_k}}} + {\varepsilon _{k,{m_k}}}} \right|}}} \right\rbrace } \right\rbrace  =& \frac{{\sqrt{\pi }\beta _k^2}}{{2\sqrt{\beta _k^2 + \delta _{k,{m_k}}^2} }}, \\\mathbb {E}\left\lbrace  {\mathcal {I}\left\lbrace  {\frac{{\left( {h_{k,{m_k}}^* + \varepsilon _{k,{m_k}}^*} \right){h_{k,{m_k}}}}}{{\left| {{h_{k,{m_k}}} + {\varepsilon _{k,{m_k}}}} \right|}}} \right\rbrace } \right\rbrace  =& 0. \) 
